A326432 - OEIS

%I #9 Jul 11 2019 20:08:37

%S 1,4,43,762,19573,672374,29390733,1578973510,101612589283,

%T 7679375658354,670906936259299,66891320576455142,7530075312966689409,

%U 948460025747139087802,132635012110499511683869,20454728573277460691412006,3458323793329321035116835859,637694404371402843143395980434,127650318560095585201739965521651

%N E.g.f.: exp(-3) * Sum_{n>=0} ((1+x)^n + 2)^n / n!.

%C More generally, the following sums are equal:

%C (1) exp(-r*(p+1)) * Sum_{n>=0} (q^n + p)^n * r^n / n!,

%C (2) exp(-r*(p+1)) * Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n / n!,

%C here, q = 1+x, p = 2, r = 1.

%F E.g.f.: exp(-3) * Sum_{n>=0} ((1+x)^n + 2)^n / n!.

%F E.g.f.: exp(-3) * Sum_{n>=0} (1+x)^(n^2) * exp( 2*(1+x)^n ) / n!.

%e E.g.f.: A(x) = 1 + 4*x + 43*x^2/2! + 762*x^3/3! + 19573*x^4/4! + 672374*x^5/5! + 29390733*x^6/6! + 1578973510*x^7/7! + 101612589283*x^8/8! + 7679375658354*x^9/9! + 670906936259299*x^10/10! + ...

%e such that

%e A(x) = exp(-3) * (1 + ((1+x) + 2) + ((1+x)^2 + 2)^2/2! + ((1+x)^3 + 2)^3/3! + ((1+x)^4 + 2)^4/4! + ((1+x)^5 + 2)^5/5! + ((1+x)^6 + 2)^6/6! + ...)

%e also,

%e A(x) = exp(-3) * (exp(2) + (1+x)*exp(2*(1+x)) + (1+x)^4*exp(2*(1+x)^2)/2! + (1+x)^9*exp(2*(1+x)^3)/3! + (1+x)^16*exp(2*(1+x)^4)/4! + (1+x)^25*exp(2*(1+x)^5)/5! + (1+x)^36*exp(2*(1+x)^6)/6! + ...).

%o (PARI) /* Requires appropriate precision */

%o \p200

%o {a(n) = my(A = exp(-3) * sum(m=0,n+300, ((1+x)^m + 2 +x*O(x^n))^m / m! )); round(n!*polcoeff(A,n))}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A326431.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 09 2019